Geometry of the Donaldson-Friedman Pushout: Twistor degenerations and instanton charge

TL;DR

This paper provides an explicit algebro-geometric model for twistor degenerations in the Donaldson-Friedman construction, analyzing the singular central fibre and its implications for bundle gluing and instanton charge.

Contribution

It introduces a detailed algebraic model for the twistor degeneration, including Chow ring computations and topological interpretations relevant to instantons.

Findings

Explicit Chow ring as an equalizer of the two branches.

Specialization formula for semistable smoothings.

Additivity results for second Chern cycle and polarized charge.

Abstract

We study the singular central fibre arising in the Donaldson-Friedman construction for twistor spaces of connected sums, viewing it as a Ferrand pushout of two blown-up twistor spaces along the exceptional quadric. This provides an explicit algebro-geometric model for the twistor degeneration associated with the connected-sum construction. We describe its operational Chow ring explicitly as an equalizer of the Chow rings of the two branches, derive a componentwise specialization formula for semistable smoothings, and obtain rigid gluing constraints for surfaces across the double locus. We then interpret the local semistable equation through the Kato-Nakayama space, identifying the fixed-phase boundary as a natural circle bundle over the exceptional quadric and relating it to the topology of the neck. Finally, motivated by the twistor description of instantons, we apply this algebro-geometric formalism to bundles arising from Ward and Hartshorne-Serre data, proving additivity results for the second Chern cycle and for the polarized charge across the pushout. In this way, the singular central fibre becomes an explicitly computable carrier of bundle gluing, logarithmic neck data, and instanton-type charge in the Donaldson-Friedman setting.